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INSTRUCTORS Carleen Eaton Grant Fraser Eric Smith
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Eric Smith

Eric Smith

Applications of Rational Expressions

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Table of Contents

I. Properties of Real Numbers
Basic Types of Numbers

30m 41s

Intro
0:00
Objectives
0:07
Basic Types of Numbers
0:36
Natural Numbers
1:02
Whole Numbers
1:29
Integers
2:04
Rational Numbers
2:38
Irrational Numbers
5:06
Imaginary Numbers
6:48
Basic Types of Numbers Cont.
8:09
The Big Picture
8:10
Real vs. Imaginary Numbers
8:30
Rational vs. Irrational Numbers
8:48
Basic Types of Numbers Cont.
10:55
Number Line
11:06
Absolute Value
11:44
Inequalities
12:39
Example 1
13:16
Example 2
17:30
Example 3
21:56
Example 4
24:27
Example 5
27:48
Operations on Numbers

19m 26s

Intro
0:00
Objectives
0:06
Operations on Numbers
0:25
Addition
0:53
Subtraction
1:33
Multiplication & Division
2:19
Exponents
3:24
Bases
4:04
Square Roots
4:59
Principle Square Roots
5:09
Perfect Squares
6:32
Simplifying and Combining Roots
6:52
Example 1
8:16
Example 2
12:30
Example 3
14:02
Example 4
16:27
Order of Operations

12m 6s

Intro
0:00
Objectives
0:06
The Order of Operations
0:25
Work Inside Parentheses
0:42
Simplify Exponents
0:52
Multiplication & Division from Left to Right
0:57
Addition & Subtraction from Left to Right
1:11
Remember PEMDAS
1:21
The Order of Operations Cont.
2:27
Example
2:43
Example 1
3:55
Example 2
5:36
Example 3
7:35
Example 4
8:56
Properties of Real Numbers

18m 52s

Intro
0:00
Objectives
0:07
The Properties of Real Numbers
0:23
Commutative Property of Addition and Multiplication
0:44
Associative Property of Addition and Multiplication
1:50
Distributive Property of Multiplication Over Addition
3:20
Division Property of Zero
4:46
Division Property of One
5:23
Multiplication Property of Zero
5:56
Multiplication Property of One
6:17
Addition Property of Zero
6:29
Why Are These Properties Important?
6:53
Example 1
9:16
Example 2
13:04
Example 3
14:30
Example 4
16:57
II. Linear Equations
The Vocabulary of Linear Equations

12m 22s

Intro
0:00
Objectives
0:09
The Vocabulary of Linear Equations
0:44
Variables
0:52
Terms
1:09
Coefficients
1:40
Like Terms
2:18
Examples of Like Terms
2:37
Expressions
4:01
Equations
4:26
Linear Equations
5:04
Solutions
5:55
Example 1
6:16
Example 2
7:16
Example 3
8:45
Example 4
10:20
Solving Linear Equations in One Variable

28m 52s

Intro
0:00
Objectives
0:08
Solving Linear Equations in One Variable
0:34
Conditional Cases
0:51
Identity Cases
1:09
Contradiction Cases
1:30
Solving Linear Equations in One Variable Cont.
2:00
Addition Property of Equality
2:10
Multiplication Property of Equality
2:43
Steps to Solve Linear Equations
3:14
Example 1
4:22
Example 2
8:21
Example 3
12:32
Example 4
14:19
Example 5
17:25
Example 6
22:17
Solving Formulas

12m 2s

Intro
0:00
Objectives
0:06
Solving Formulas
0:18
Formulas
0:26
Use the Same Properties as Solving Linear Equations
1:36
Addition Property of Equality
1:55
Multiplication Property of Equality
1:58
Steps to Solve Formulas
2:43
Example 1
3:56
Example 2
6:09
Example 3
8:39
Applications of Linear Equations

28m 41s

Intro
0:00
Objectives
0:10
Applications of Linear Equations
0:43
The Six-Step Method to Solving Word Problems
0:55
Common Terms
3:12
Example 1
5:03
Example 2
9:40
Example 3
13:48
Example 4
17:58
Example 5
23:28
Applications of Linear Equations, Motion & Mixtures

24m 26s

Intro
0:00
Objectives
0:21
Motion and Mixtures
0:46
Motion Problems: Distance, Rate, and Time
1:06
Mixture Problems: Amount, Percent, and Total
1:27
The Table Method
1:58
The Beaker Method
3:38
Example 1
5:05
Example 2
9:44
Example 3
14:20
Example 4
19:13
III. Graphing
Rectangular Coordinate System

22m 55s

Intro
0:00
Objectives
0:11
The Rectangular Coordinate System
0:39
The Cartesian Coordinate System
0:40
X-Axis
0:54
Y-Axis
1:04
Origin
1:11
Quadrants
1:26
Ordered Pairs
2:10
Example 1
2:55
The Rectangular Coordinate System Cont.
6:09
X-Intercept
6:45
Y-Intercept
6:55
Relation of X-Values and Y-Values
7:30
Example 2
11:03
Example 3
12:13
Example 4
14:10
Example 5
18:38
Slope & Graphing

27m 58s

Intro
0:00
Objectives
0:11
Slope and Graphing
0:48
Standard Form
1:14
Example 1
2:24
Slope and Graphing Cont.
4:58
Slope, m
5:07
Slope is Rise over Run
6:11
Don't Mix Up the Coordinates
8:20
Example 2
9:39
Slope and Graphing Cont.
14:26
Slope-Intercept Form
14:34
Example 3
16:55
Example 4
18:00
Slope and Graphing Cont.
19:00
Rewriting an Equation in Slope-Intercept Form
19:39
Rewriting an Equation in Standard Form
20:09
Slopes of Vertical & Horizontal Lines
20:56
Example 5
22:49
Example 6
24:09
Example 7
25:59
Example 8
26:57
Linear Equations in Two Variables

20m 36s

Intro
0:00
Objectives
0:13
Linear Equations in Two Variables
0:36
Point-Slope Form
1:07
Substitute in the Point and the Slope
2:21
Parallel Lines: Two Lines with the Same Slope
4:05
Perpendicular Lines: Slopes are Negative Reciprocals of Each Other
4:39
Perpendicular Lines: Product of Slopes is -1
5:24
Example 1
6:02
Example 2
7:50
Example 3
10:49
Example 4
13:26
Example 5
15:30
Example 6
17:43
IV. Functions
Introduction to Functions

21m 24s

Intro
0:00
Objectives
0:07
Introduction to Functions
0:58
Relations
1:03
Functions
1:37
Independent Variables
2:00
Dependent Variables
2:11
Function Notation
2:21
Function
3:43
Input and Output
3:53
Introduction to Functions Cont.
4:45
Domain
4:46
Range
4:55
Functions Represented by a Diagram
6:41
Natural Domain
9:11
Evaluating Functions
12:02
Example 1
13:13
Example 2
15:03
Example 3
16:18
Example 4
19:54
Graphing Functions

16m 12s

Intro
0:00
Objectives
0:09
Graphing Functions
0:54
Using Slope-Intercept Form
1:56
Vertical Line Test
2:58
Determining the Domain
4:20
Determining the Range
5:43
Example 1
6:06
Example 2
7:18
Example 3
8:31
Example 4
11:04
V. Systems of Linear Equations
Systems of Linear Equations

25m 54s

Intro
0:00
Objectives
0:13
Systems of Linear Equations
0:46
System of Equations
0:51
System of Linear Equations
1:15
Solutions
1:35
Points as Solutions
1:53
Finding Solutions Graphically
5:13
Example 1
6:37
Example 2
12:07
Systems of Linear Equations Cont.
17:01
One Solution, No Solution, or Infinite Solutions
17:10
Example 3
18:31
Example 4
22:37
Solving a System Using Substitution

20m 1s

Intro
0:00
Objectives
0:09
Solving a System Using Substitution
0:32
Substitution Method
1:24
Substitution Example
2:35
One Solution, No Solution, or Infinite Solutions
7:50
Example 1
9:45
Example 2
12:48
Example 3
15:01
Example 4
17:30
Solving a System Using Elimination

19m 40s

Intro
0:00
Objectives
0:09
Solving a System Using Elimination
0:27
Elimination Method
0:42
Elimination Example
2:01
One Solution, No Solution, or Infinite Solutions
7:05
Example 1
8:53
Example 2
11:46
Example 3
15:37
Example 4
17:45
Applications of Systems of Equations

24m 34s

Intro
0:00
Objectives
0:12
Applications of Systems of Equations
0:30
Word Problems
1:31
Example 1
2:17
Example 2
7:55
Example 3
13:07
Example 4
17:15
VI. Inequalities
Solving Linear Inequalities in One Variable

17m 13s

Intro
0:00
Objectives
0:08
Solving Linear Inequalities in One Variable
0:37
Inequality Expressions
0:46
Linear Inequality Solution Notations
3:40
Inequalities
3:51
Interval Notation
4:04
Number Lines
4:43
Set Builder Notation
5:24
Use Same Techniques as Solving Equations
6:59
'Flip' the Sign when Multiplying or Dividing by a Negative Number
7:12
'Flip' Example
7:50
Example 1
8:54
Example 2
11:40
Example 3
14:01
Compound Inequalities

16m 13s

Intro
0:00
Objectives
0:07
Compound Inequalities
0:37
'And' vs. 'Or'
0:44
'And'
3:24
'Or'
3:35
'And' Symbol, or Intersection
3:51
'Or' Symbol, or Union
4:13
Inequalities
4:41
Example 1
6:22
Example 2
9:30
Example 3
11:27
Example 4
13:49
Solving Equations with Absolute Values

14m 12s

Intro
0:00
Objectives
0:08
Solve Equations with Absolute Values
0:18
Solve Equations with Absolute Values Cont.
1:11
Steps to Solving Equations with Absolute Values
2:21
Example 1
3:23
Example 2
6:34
Example 3
10:12
Inequalities with Absolute Values

17m 7s

Intro
0:00
Objectives
0:07
Inequalities with Absolute Values
0:23
Recall…
2:08
Example 1
3:39
Example 2
6:06
Example 3
8:14
Example 4
10:29
Example 5
13:29
Graphing Inequalities in Two Variables

15m 33s

Intro
0:00
Objectives
0:07
Graphing Inequalities in Two Variables
0:32
Split Graph into Two Regions
1:53
Graphing Inequalities
5:44
Test Points
6:20
Example 1
7:11
Example 2
10:17
Example 3
13:06
Systems of Inequalities

21m 13s

Intro
0:00
Objectives
0:08
Systems of Inequalities
0:24
Test Points
1:10
Steps to Solve Systems of Inequalities
1:25
Example 1
2:23
Example 2
7:28
Example 3
12:51
VII. Polynomials
Integer Exponents

44m 51s

Intro
0:00
Objectives
0:09
Integer Exponents
0:42
Exponents 'Package' Multiplication
1:25
Example 1
2:00
Example 2
3:13
Integer Exponents Cont.
4:50
Product Rule for Exponents
4:51
Example 3
7:16
Example 4
10:15
Integer Exponents Cont.
13:13
Power Rule for Exponents
13:14
Power Rule with Multiplication and Division
15:33
Example 5
16:18
Integer Exponents Cont.
20:04
Example 6
20:41
Integer Exponents Cont.
25:52
Zero Exponent Rule
25:53
Quotient Rule
28:24
Negative Exponents
30:14
Negative Exponent Rule
32:27
Example 7
34:05
Example 8
36:15
Example 9
39:33
Example 10
43:16
Adding & Subtracting Polynomials

18m 33s

Intro
0:00
Objectives
0:07
Adding and Subtracting Polynomials
0:25
Terms
0:33
Coefficients
0:51
Leading Coefficients
1:13
Like Terms
1:29
Polynomials
2:21
Monomials, Binomials, Trinomials, and Polynomials
5:41
Degrees
7:00
Evaluating Polynomials
8:12
Adding and Subtracting Polynomials Cont.
9:25
Example 1
11:48
Example 2
13:00
Example 3
14:41
Example 4
16:15
Multiplying Polynomials

25m 7s

Intro
0:00
Objectives
0:06
Multiplying Polynomials
0:41
Distributive Property
1:00
Example 1
2:49
Multiplying Polynomials Cont.
8:22
Organize Terms with a Table
8:23
Example 2
13:40
Multiplying Polynomials Cont.
16:33
Multiplying Binomials with FOIL
16:48
Example 3
18:49
Example 4
20:04
Example 5
21:42
Dividing Polynomials

44m 56s

Intro
0:00
Objectives
0:07
Dividing Polynomials
0:29
Dividing Polynomials by Monomials
2:10
Dividing Polynomials by Polynomials
2:59
Dividing Numbers
4:09
Dividing Polynomials Example
8:39
Example 1
12:35
Example 2
14:40
Example 3
16:45
Example 4
21:13
Example 5
24:33
Example 6
29:02
Dividing Polynomials with Synthetic Division Method
33:36
Example 7
38:43
Example 8
42:24
VIII. Factoring Polynomials
Greatest Common Factor & Factor by Grouping

28m 27s

Intro
0:00
Objectives
0:09
Greatest Common Factor
0:31
Factoring
0:40
Greatest Common Factor (GCF)
1:48
GCF for Polynomials
3:28
Factoring Polynomials
6:45
Prime
8:21
Example 1
9:14
Factor by Grouping
14:30
Steps to Factor by Grouping
17:03
Example 2
17:43
Example 3
19:20
Example 4
20:41
Example 5
22:29
Example 6
26:11
Factoring Trinomials

21m 44s

Intro
0:00
Objectives
0:06
Factoring Trinomials
0:25
Recall FOIL
0:26
Factor a Trinomial by Reversing FOIL
1:52
Tips when Using Reverse FOIL
5:31
Example 1
7:04
Example 2
9:09
Example 3
11:15
Example 4
13:41
Factoring Trinomials Cont.
15:50
Example 5
18:42
Factoring Trinomials Using the AC Method

30m 9s

Intro
0:00
Objectives
0:08
Factoring Trinomials Using the AC Method
0:27
Factoring when Leading Term has Coefficient Other Than 1
1:07
Reversing FOIL
1:18
Example 1
1:46
Example 2
4:28
Factoring Trinomials Using the AC Method Cont.
7:45
The AC Method
8:03
Steps to Using the AC Method
8:19
Tips on Using the AC Method
9:29
Example 3
10:45
Example 4
16:50
Example 5
21:08
Example 6
24:58
Special Factoring Techniques

30m 14s

Intro
0:00
Objectives
0:07
Special Factoring Techniques
0:26
Difference of Squares
1:46
Perfect Square Trinomials
2:38
No Sum of Squares
3:32
Special Factoring Techniques Cont.
4:03
Difference of Squares Example
4:04
Perfect Square Trinomials Example
5:29
Example 1
7:31
Example 2
9:59
Example 3
11:47
Example 4
15:09
Special Factoring Techniques Cont.
19:07
Sum of Cubes and Difference of Cubes
19:08
Example 5
23:13
Example 6
26:12
IX. Quadratic Equations
Solving Quadratic Equations by Factoring

23m 38s

Intro
0:00
Objectives
0:08
Solving Quadratic Equations by Factoring
0:19
Quadratic Equations
0:20
Zero Factor Property
1:39
Zero Factor Property Example
2:34
Example 1
4:00
Solving Quadratic Equations by Factoring Cont.
5:54
Example 2
7:28
Example 3
11:09
Example 4
14:22
Solving Quadratic Equations by Factoring Cont.
18:17
Higher Degree Polynomial Equations
18:18
Example 5
20:22
Solving Quadratic Equations

29m 27s

Intro
0:00
Objectives
0:12
Solving Quadratic Equations
0:29
Linear Factors
0:38
Not All Quadratics Factor Easily
1:22
Principle of Square Roots
3:36
Completing the Square
4:50
Steps for Using Completing the Square
5:15
Completing the Square Works on All Quadratic Equations
6:41
The Quadratic Formula
7:28
Discriminants
8:25
Solving Quadratic Equations - Summary
10:11
Example 1
11:54
Example 2
13:03
Example 3
16:30
Example 4
21:29
Example 5
25:07
Equations in Quadratic Form

16m 47s

Intro
0:00
Objectives
0:08
Equations in Quadratic Form
0:24
Using a Substitution
0:53
U-Substitution
1:26
Example 1
2:07
Example 2
5:36
Example 3
8:31
Example 4
11:14
Quadratic Formulas & Applications

29m 4s

Intro
0:00
Objectives
0:09
Quadratic Formulas and Applications
0:35
Squared Variable
0:40
Principle of Square Roots
0:51
Example 1
1:09
Example 2
2:04
Quadratic Formulas and Applications Cont.
3:34
Example 3
4:42
Example 4
13:33
Example 5
20:50
Graphs of Quadratics

26m 53s

Intro
0:00
Objectives
0:06
Graphs of Quadratics
0:39
Axis of Symmetry
1:46
Vertex
2:12
Transformations
2:57
Graphing in Quadratic Standard Form
3:23
Example 1
5:06
Example 2
6:02
Example 3
9:07
Graphs of Quadratics Cont.
11:26
Completing the Square
12:02
Vertex Shortcut
12:16
Example 4
13:49
Example 5
17:25
Example 6
20:07
Example 7
23:43
Polynomial Inequalities

21m 42s

Intro
0:00
Objectives
0:07
Polynomial Inequalities
0:30
Solving Polynomial Inequalities
1:20
Example 1
2:45
Polynomial Inequalities Cont.
5:12
Larger Polynomials
5:13
Positive or Negative Intervals
7:16
Example 2
9:01
Example 3
13:53
X. Rational Equations
Multiply & Divide Rational Expressions

26m 41s

Intro
0:00
Objectives
0:09
Multiply and Divide Rational Expressions
0:44
Rational Numbers
0:55
Dividing by Zero
1:45
Canceling Extra Factors
2:43
Negative Signs in Fractions
4:52
Multiplying Fractions
6:26
Dividing Fractions
7:17
Example 1
8:04
Example 2
14:01
Example 3
16:23
Example 4
18:56
Example 5
22:43
Adding & Subtracting Rational Expressions

20m 24s

Intro
0:00
Objectives
0:07
Adding and Subtracting Rational Expressions
0:41
Common Denominators
0:52
Common Denominator Examples
1:14
Steps to Adding and Subtracting Rational Expressions
2:39
Example 1
3:34
Example 2
5:27
Adding and Subtracting Rational Expressions Cont.
6:57
Least Common Denominators
6:58
Transitioning from Fractions to Rational Expressions
9:08
Identifying Least Common Denominators for Rational Expressions
9:56
Subtracting vs. Adding
10:41
Example 3
11:19
Example 4
12:36
Example 5
15:08
Example 6
16:46
Complex Fractions

18m 23s

Intro
0:00
Objectives
0:09
Complex Fractions
0:37
Dividing to Simplify Complex Fractions
1:10
Example 1
2:03
Example 2
3:58
Complex Fractions Cont.
9:15
Using the Least Common Denominator to Simplify Complex Fractions
9:16
Both Methods Lead to the Same Answer
10:07
Example 3
10:42
Example 4
14:28
Solving Rational Equations

16m 24s

Intro
0:00
Objectives
0:07
Solving Rational Equations
0:23
Isolate the Specified Variable
1:23
Example 1
1:58
Example 2
5:00
Example 3
8:23
Example 4
13:25
Rational Inequalities

18m 54s

Intro
0:00
Objectives
0:06
Rational Inequalities
0:18
Testing Intervals for Rational Inequalities
0:38
Steps to Solving Rational Inequalities
1:05
Tips to Solving Rational Inequalities
2:27
Example 1
3:33
Example 2
12:21
Applications of Rational Expressions

20m 20s

Intro
0:00
Objectives
0:07
Applications of Rational Expressions
0:27
Work Problems
1:05
Example 1
2:58
Example 2
6:45
Example 3
13:17
Example 4
16:37
Variation & Proportion

27m 4s

Intro
0:00
Objectives
0:10
Variation and Proportion
0:34
Variation
0:35
Inverse Variation
1:01
Direct Variation
1:10
Setting Up Proportions
1:31
Example 1
2:27
Example 2
5:36
Variation and Proportion Cont.
8:29
Inverse Variation
8:30
Example 3
9:20
Variation and Proportion Cont.
12:41
Constant of Proportionality
12:42
Example 4
13:59
Variation and Proportion Cont.
16:17
Varies Directly as the nth Power
16:30
Varies Inversely as the nth Power
16:53
Varies Jointly
17:09
Combining Variation Models
17:36
Example 5
19:09
Example 6
22:10
XI. Radical Equations
Rational Exponents

14m 32s

Intro
0:00
Objectives
0:07
Rational Exponents
0:32
Power on Top, Root on Bottom
1:05
Example 1
1:37
Rational Exponents Cont.
4:04
Using Rules from Exponents for Radicals as Exponents
4:05
Combining Terms Under a Single Root
4:50
Example 2
5:21
Example 3
7:39
Example 4
11:23
Example 5
13:14
Simplify Rational Exponents

15m 12s

Intro
0:00
Objectives
0:07
Simplify Rational Exponents
0:25
Product Rule for Radicals
0:26
Product Rule to Simplify Square Roots
1:11
Quotient Rule for Radicals
1:42
Applications of Product and Quotient Rules
2:17
Higher Roots
2:48
Example 1
3:39
Example 2
6:35
Example 3
8:41
Example 4
11:09
Adding & Subtracting Radicals

17m 22s

Intro
0:00
Objectives
0:07
Adding and Subtracting Radicals
0:33
Like Terms
1:29
Bases and Exponents May be Different
2:02
Bases and Powers Must be Same when Adding and Subtracting
2:42
Add Radicals' Coefficients
3:55
Example 1
4:47
Example 2
6:00
Adding and Subtracting Radicals Cont.
7:10
Simplify the Bases to Look the Same
7:25
Example 3
8:23
Example 4
11:45
Example 5
15:10
Multiply & Divide Radicals

19m 24s

Intro
0:00
Objectives
0:08
Multiply and Divide Radicals
0:25
Rules for Working With Radicals
0:26
Using FOIL for Radicals
1:11
Don’t Distribute Powers
2:54
Dividing Radical Expressions
4:25
Rationalizing Denominators
6:40
Example 1
7:22
Example 2
8:32
Multiply and Divide Radicals Cont.
9:23
Rationalizing Denominators with Higher Roots
9:25
Example 3
10:51
Example 4
11:53
Multiply and Divide Radicals Cont.
13:13
Rationalizing Denominators with Conjugates
13:14
Example 5
15:52
Example 6
17:25
Solving Radical Equations

15m 5s

Intro
0:00
Objectives
0:07
Solving Radical Equations
0:17
Radical Equations
0:18
Isolate the Roots and Raise to Power
0:34
Example 1
1:13
Example 2
3:09
Solving Radical Equations Cont.
7:04
Solving Radical Equations with More than One Radical
7:05
Example 3
7:54
Example 4
13:07
Complex Numbers

29m 16s

Intro
0:00
Objectives
0:06
Complex Numbers
1:05
Imaginary Numbers
1:08
Complex Numbers
2:27
Real Parts
2:48
Imaginary Parts
2:51
Commutative, Associative, and Distributive Properties
3:35
Adding and Subtracting Complex Numbers
4:04
Multiplying Complex Numbers
6:16
Dividing Complex Numbers
8:59
Complex Conjugate
9:07
Simplifying Powers of i
14:34
Shortcut for Simplifying Powers of i
18:33
Example 1
21:14
Example 2
22:15
Example 3
23:38
Example 4
26:33
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Applications of Rational Expressions

  • When working on a word problem the unknown could end up in the denominator. These often lead to rational expression.
  • Some common word problems that lead to rational expressions are ones that involve motion or work.
  • If a job can be completed in t units of time, then the rate of work is given by 1/t.
  • By multiplying the rate by the amount of time, we can figure out how much of a job has been completed at any given time.
  • From working with motion we know that distance = rate x time. This leads to other equations such as rate = distance/time and time = distance/rate

Applications of Rational Expressions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Objectives 0:07
  • Applications of Rational Expressions 0:27
    • Work Problems
  • Example 1 2:58
  • Example 2 6:45
  • Example 3 13:17
  • Example 4 16:37

Transcription: Applications of Rational Expressions

Welcome back to www.educator.com.0000

In this lesson we are going to take a look at applications of rational equations.0002

In some of the examples I have cooked up we will look at how some examples involves a just numbers.0009

There are ones that involve motion and some of my favorite ones that involve work.0015

This one is going to be a little bit trickier to think about, but once you get the process done0020

you will see that the work problems are not so bad.0024

Depending on the unknowns in the problem and depending on how we should go ahead and package everything up0031

there could be a few situations that actually lead to rational expressions.0037

Think about those fractions or where an unknown ends up on the denominator.0041

What we want to do is be able to solve these using many of the techniques that we have picked up for rational equations0050

and see how we can recognize these in various different situations, such as numbers, motion, and especially work.0056

The work problems are kind of unique and that you want to know how to represent the situation.0067

If you know how long it takes to complete an entire job then you know the rate of work is given by the following formula 1/T.0075

The way you can read this is by using that T for the amount it takes to do that one job.0084

Here is a quick example.0090

Let us say it takes Betty 7 hours to paint her entire room.0092

Well, that means that every single hour 1/7 of the room is going to be painted.0098

We are going to make a little chart for just keeping track of everything.0103

Maybe x will be my time and let us say how much of the room has been painted so far.0109

One hour, two hours, three hours and make sure you jump all the way to 7.0120

After she is in the room for 7 hours, she will have the entire room painted, the whole thing.0127

If you scale it back, what if she is only working for one hour then only 1/7 of the room is painted.0135

If she is in there for two hours, 2/7 of the room is painted and if she is in there for three 3/7.0144

You can see that we are just incrementing this thing by exactly 1/7 every time.0151

We can make some adjustments to our formula here and say that T would be the amount it takes to do that one for the person0156

and may be multiply it by x, that would represent how long they have been doing that particular job.0165

Watch for that to play a key component with our work problem in just a bit.0173

Let us first see our example of numbers and just see something where our variable ends up in the denominator.0180

In this one we have a certain number and we are going to add it to the numerator and subtract it from the denominator of 7/3.0186

The result equals the reciprocal of 7/3 and we are interested in finding that number.0194

Let us first write down our unknown.0202

x is the unknown number.0205

We construct a model situation here.0217

Take that unknown number and add it to the numerator, but subtract it from the denominator of 7/3.0219

Here is 7/3, so we are adding it to the numerator and subtract it from the denominator.0227

The result equals the reciprocal of 7/3, that is like 7/3 but we flipped it over.0238

This will be our rational inequality here.0245

What we have to do is work on solving it.0249

To solve many of our rational equalities we work on finding a least common denominator0253

which I can see for this one will be 3 - x and 7.0261

Let us give that missing piece to each of the fractions.0268

Here is my original (x + x) (3 – x) = 3/7 let us use some extra space in here.0272

The one on the left, it could use 7, let us put that in there.0286

The one on the right it is missing the 3 – x.0294

At that point, the denominators will be exactly the same.0302

We will just go ahead and focus on the tops of each of these.0307

7 × 7 + x = 3, 3 – x.0312

Continuing on and solving for our x we will go ahead and distribute our 7 and 3.0321

That will give us 49 7x = 9 - 3x.0328

Moving along pretty good.0336

Let us go ahead and add 3x to both sides giving us a 49 + 10x = 9.0338

We will go ahead and subtract 49 from both sides.0351

I have x = 10x is equal to -40.0359

Dividing both sides by 10, I have that x = -4.0368

Just like when we are working with equations, it has to make sense in our original.0374

Looking at the original rational expression I have a restricted value of 3 and I know that it is not 3 that will make the bottom 0.0379

I do not get any restricted values from the other faction because it is simply always 7 on the bottom.0388

Since the -4 is not 3, I'm going to keep it as a valid solution.0394

That one looks good.0402

Let us look at one that involves motion.0407

We will set these up using a table and also use that same idea to help us organize this information.0409

A boat can go 10 miles against a current in the same time it can go 30 miles with the current.0416

The current flows at 4 mph, find the speed of the boat with no current.0422

We have an interesting situation.0428

We have a boat that looks something like this and we have the flow of the river.0430

Now in one situation, it is fighting against the current, and the way you want to think of that in relation to its speed0439

is that the speed of the river is taking away some of the speed of the boat.0447

You will see a subtraction process.0452

If the boat is going in the same direction of the river, they are both helping each other out0456

and you will see an addition problem with both of their speed.0461

You will know they are both helping each other out.0464

Let us see if we organize this information so I can see how to connect it.0468

We need to think of two different situations.0492

We are either going against, or we are going with the river.0495

We will look at the rate, the time, and the distance.0502

This will help us keep track of everything.0511

And of course we are leaving unknown in here.0513

Since we are finding the speed of the boat with no current, let us set that as our unknown.0516

x is the speed of the boat with no current.0522

I think we have a good set up and we can start organizing our information.0540

In the first bit of this problem we know that it can go 10 miles where it is going against the current.0545

That is its distance.0553

It went 10 miles when it is going against the current.0554

It can do that in the same time it can go 30 miles with the current.0558

A little bit of different information, this one will be 30 when it is going that way.0564

The current of the river flows at 4 mph.0570

If we are looking at the speed of the boat and it is going against that river, probably this will be the boat - the current.0575

If we are looking at it going with the river, that will be the speed of the boat + the speed of the current.0585

They are helping each other out.0590

The only thing we do not know in here is the time, but I do know that the time was exactly the same for both of these situations.0593

Let us see what do we got here.0603

I will go ahead and create an equation for each of these.0604

x -4 × time = 10 and x + 4 × time = 30.0607

I know the times are exactly the same for each of these, let us solve them both for time.0620

This one I will go ahead and divide both sides by x - 4.0628

In this one I will divide by x + 4.0631

And I'm ready to develop that rational equation.0638

I will set each of these equal to each other since the times are equal to each other.0641

I have a rational equation then I can go ahead and try and solve.0651

10 ÷ x – 4 = 30 ÷ x + 40655

To get through solving process, we find our common denominator.0661

I'm going to give x + 4 on the left side here and I will give x - 4 to the other side.0668

We will note that we made the denominators exactly the same.0687

We just need to focus on the tops of each of these.0691

Continuing on, you will distribute 10 and we will distribute 30.0702

10x + 40 = 30x -1200710

Subtracting a 10x from both sides will give us 40 = 20x -120 and let us go ahead and add 120 to both sides.0720

160 = 200739

We can divide both sides by 20 and I have that x is equal to 8.0745

Let us make sure that it makes sense.0754

Some restricted values I have for my equations here I know that x cannot equal 4 and -40757

and fortunately both that we have found for a possible solution is neither of those.0766

We can say the speed of the boat in still water would be 8 mph and then this guy is done.0772

Do not be afraid to use those tables from earlier to organize your information.0790

Joe and Steve operated a small roofing company and Mario can roof an average house alone in 9 hours.0800

Al can roof a house alone in 8 hours.0809

We want to know how long will it take them to do their job if they work together.0812

First we need to figure out the rates of each of them individually.0818

Let us go ahead and focus on Joe.0821

Joe can roof an average house alone in 9 hours.0830

Looking at just Joe we know that every hour he will get 1/9 of that house done.0834

Steve over here can roof the house in 8 hours.0844

He is working every single hour 1/8 of that house will be done.0853

We can put in that time on it.0858

Let x be the number of hours.0861

You have 1/9 × however many hours they work and 1/8 × qualified by however many hours Steve works.0871

We want to know how long it will take them to do the job if they work together.0881

We will take each of their work that they are doing and we will add them since they are working together,0889

we want to know when they will complete one job.0895

We have all of our information here and we can go ahead and try and solve this.0899

Our LCD would be 72.0906

We will multiply that through on all parts.0911

Doing some reducing 72 and 9 = 8, 72 and 8 = 9 and now we have an equation 8x + 9x = 72.0923

Combining together what we have on the left side this would be 17x is equal to 720942

and we can divide both sides of that by 17 to get x = 72 ÷ 17.0951

It is looking pretty good.0959

That represents how long it will take them to work together.0961

If you want to represent that as a decimal, you could go ahead and take 72 ÷ 17.0964

When I did that I got about 4.24 hours, I did round it.0971

That gives me a better idea of how long it took them when working together.0980

Anytime when you are working with these types of problems, it should be less than any one of them working by themselves,0985

since they are helping each other out.0992

This one looks good.0995

In this last example we are going to look at a water tank that has two hoses connected to it.0998

Even though this is not a work problem you can see that we can set it up in much the same way.1003

Let us go ahead and give it a read.1008

The information that we have is that the first hose can fill the entire tank in 5 hours.1011

The second hose connected to this tank it can empty it in 3 hours.1018

If we start with a completely full tank and then we turned both of them open,1023

the question is how long will it take it to empty the entire tank?1029

Let us have an accrued picture of what we are dealing with here.1034

This would be our water tank and we have one hose that is going in and one hose that is going out.1039

The hose that is going in, it can fill the tank in 5 hours.1054

That means if we just leave it on every hour 1/5 of that tank will fill up.1058

I know its rate is 1/5.1063

If I look at emptying the tank it is a much smaller time to empty it, 1/3.1068

If I did have them both open I would know that the second one1076

would be able to empty the tank since it empties faster than the first one can fill it.1080

Let us see what that we can do to set this one out.1086

The first thing I want to consider was how much the tank is going to be empty every single hour.1091

Starting with the 1/3 I know that every hour that passes by the 1/3 of it will be emptied out.1097

x is the number of hours.1106

The 1/5 coming in is not emptying the tank but it is actually filling it back up.1118

We will say it is the opposite of emptying the tank by 1/5 and it will do that for every hour.1125

We want to know is when will one tank be completely empty.1132

We have a lot of similar components for this one and it often look like a lot of our work problems.1137

All we have to do is go ahead and solve it.1145

Our LCD here that would be 15.1148

Let us multiply all parts by 15 and see what that does.1153

15, 15 and 15.1159

Canceling out some extra stuff here I get 5x - 3x = 15.1163

Doing a little bit of combining on the left side I have 2x =15 or x = 15 ÷ 2 or 7 ½ hours.1173

Notice how in this case it is taking longer since both of them are open.1190

The reason for this is that they are not working with each other.1194

They are working against each other.1198

One was trying to fill the tank, and one is trying to empty the tank.1199

Use those bits of information you can find along the word problem to give you a little bit of intuition about your final solution.1205

In that way you can be assured that it does make sense in the context of the problem.1212

Thank you for watching www.educator.com.1217

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